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Mathematics 9 Algebra g n g olvi Number onin S s m ea e R cy ProPblroGbeaSobtmailtieittsyrtiycsstanding Fluen Under Measurement National Curriculum Dr Terry Dwyer Mathematics 9 National Curriculum Dr Terry Dwyer CertT, BAppSc, BEd, GDipEd, MEd(Hons), PhD Head of Mathematics www.drdwyer.com.au The cover: Maths is the pathway to success in our technological economy and society. “In today’s world, economic access and full citizenship depend crucially on math and science literacy.” - Robert P. Moses. “Mathematics makes a significant contribution to our modern society; its basic skills are vital for the life opportunities of the youth. Let us all embrace and encourage the study of this indispensable subject.“ - Alfred L. Teye. Dr Dwyer Pty Ltd ABN 27105593922 10 Moss Court Stanthorpe 4380 www.drdwyer.com.au Copyright © Dr Dwyer Pty Ltd First Published 2011 COPYRIGHT Apart from fair dealing for the purposes of study, research, criticism or review, or as permitted under Part VB of the Copyright Act, no part of this book may be reproduced by any process without permission. National Library of Australia Cataloguing-in-Publication entry Dwyer, T. (Terry), Mathematics 9 : National Curriculum 1st ed. Includes index. For secondary school year age. ISBN: 978-0-646-55957-5 1. Mathematics--Problems, exercises etc. 510 Published in Australia by Dr Dwyer Pty Ltd Printed in Malaysia Contents - Overview Chapter 1 Indices 1 1 Chapter 2 Algebra 1 17 Chapter 3 Area 33 Chapter 4 Linear & Non-linear Graphs 49 Chapter 5 Review 1 65 Chapter 6 Proportion 73 Chapter 7 Pythagoras' Theorem 89 Chapter 8 Geometry 103 Chapter 9 Statistics 119 Chapter 10 Review 2 135 Chapter 11 Indices 2 143 Chapter 12 Trigonometry 1 159 Chapter 13 Volume 175 Chapter 14 Probability 189 Chapter 15 Review 3 205 Chapter 16 Coordinate Geometry 213 Chapter 17 Trigonometry 2 229 Chapter 18 Algebra 2 245 Chapter 19 Data 261 Chapter 20 Review 4 275 Glossary 285 Answers 295 Index 311 Chapter 1 Indices 1 1 Chapter 3 Area 33 A Task 1 A Task 33 History 1 History 33 WarmUp 2 Area Warmup 34 Indices 3 Composite Shapes 35 Index Law 1 4 Prisms 36 Index Law 2 4 Surface Area 37 Index Law 3 5 Cylinders 38 Index Law 4 5 Circle Review 38 Index Law 5 6 Cylinder Surface Area 39 Summary 8 All Together 40 Mental Computation 9 Mental Computation 41 NAPLAN Questions 10 NAPLAN Questions 42 Competition Questions 11 Competition Questions 43 Investigations 12 Investigations 44 A Couple of Puzzles 13 Technology 45 A Game 13 A Couple of Puzzles 46 A Sweet Trick 13 A Game 46 Technology 14 A Sweet Trick 46 Chapter Review 1 15 Chapter Review 1 47 Chapter Review 2 16 Chapter Review 2 48 History 49 Chapter 2 Algebra 1 17 Chapter 4 Linear & Non-linear 49 A Task 17 A Task 49 History 17 Linear Rules 50 Algebra Warmup 18 Sketching Linear Graphs 53 Distributive Law 20 Linear Graphs 54 Factorisation 23 Non-Linear Graphs 55 Mental Computation 25 Mental Computation 57 NAPLAN Questions 26 NAPLAN Questions 58 Competition Questions 27 Competition Questions 59 Investigations 28 Investigations 60 A Couple of Puzzles 29 Technology 61 A Game 29 A Couple of Puzzles 62 A Sweet Trick 29 A Game 62 Technology 30 A Sweet Trick 62 Chapter Review 1 31 Chapter Review 1 63 Chapter Review 2 32 Chapter Review 2 64 Chapter 5 Review 1 65 Review 1 66 Review 2 70 Chapter 6 Proportion 73 Chapter 8 Geometry 103 A Task 73 A Task 103 History 73 History 103 Warmup 74 Congruent Triangles 104 Proportion 76 Tests for Congruent Triangles 105 Direct Proportion 78 Similarity Transformation 106 Inverse Proportion 80 Similar Triangles 108 Money and Proportion 81 Tests for Similar Triangles 109 Mental Computation 82 Mental Computation 112 Competition Questions 83 Competition Questions 113 Investigations 84 Technology 114 A Couple of Puzzles 85 A Couple of Puzzles 116 A Game 85 A Game 116 A Sweet Trick 85 A Sweet Trick 116 Technology 86 Chapter Review 1 117 Chapter Review 1 87 Chapter Review 2 118 Chapter Review 2 88 History 119 History 89 Chapter 7 Pythagoras' Theorem 89 Chapter 9 Statistics 119 A Task 89 A Task 119 Pythagorean Triads 90 Descriptive Statistics 120 Pythagoras' Theorem 91 Stem-and-Leaf Plots 122 Hypotenuse 92 Histograms 124 The Shorter Sides 93 Comparative Analysis 126 Length of a line 94 Mental Computation 128 Our Number System 95 Competition Questions 129 Mental Computation 96 Investigations 130 Competition Questions 97 A Couple of Puzzles 131 Investigations 98 A Game 131 Technology 99 A Sweet Trick 131 A Couple of Puzzles 100 Technology 132 A Game 100 Chapter Review 1 133 A Sweet Trick 100 Chapter Review 2 134 Chapter Review 1 101 Chapter Review 2 102 Chapter 10 Review 2 135 Review 1 136 Review 2 139 Chapter 11 Indices 2 143 Chapter 13 Volume 175 A Task 143 A Task 175 History 143 History 175 Warmup 144 Area Warmup 176 Index Law 1 145 Composite Shapes 177 Index Law 2 145 Prisms 178 Index Law 3 146 Units of Volume 178 Index Law 4 146 Volume of Prisms 179 Index Law 5 147 Composite Solids 180 Scientific Notation 148 Practical Applications 181 Mental Computation 152 Mental Computation 182 Competition Questions 153 Competition Questions 183 Investigations 154 Investigations 184 A Couple of Puzzles 155 A Couple of Puzzles 185 A Game 155 A Game 185 A Sweet Trick 155 A Sweet Trick 185 Technology 156 Technology 186 Chapter Review 1 157 Chapter Review 1 187 Chapter Review 2 158 Chapter Review 2 188 Chapter 12 Trigonometry 1 159 Chapter 14 Probability 189 A Task 159 A Task 189 History 159 History 189 Pythagoras' Theorem 160 Warm Up 190 Naming Sides 162 Theoretical Probability 191 Trigonometry 163 Experimental Probability 192 The Tan Ratio 164 Venn Diagrams 197 Mental Computation 168 Mental Computation 198 Competition Questions 169 Competition Questions 199 A Couple of Puzzles 170 Investigations 200 A Game 170 Technology 201 A Sweet Trick 170 A Couple of Puzzles 202 Investigations 171 A Game 202 Technology 172 A Sweet Trick 202 Chapter Review 1 173 Chapter Review 1 203 Chapter Review 2 174 Chapter Review 2 204 Chapter 15 Review 3 205 Review 1 206 Review 2 209 Chapter 16 Coordinate Geometry2 13 Chapter 18 Algebra 2 245 A Task 213 A Task 245 History 213 History 245 Distance Between 2 Points 214 Integer Warmup 246 Midpoint 216 Index Law Warmup 247 Gradient 218 Algebra Warmup 248 Formulas 220 Distributive Law 250 Mental Computation 222 Factorisation 251 Competition Questions 223 Mental Computation 254 Investigations 224 Competition Questions 255 A Couple of Puzzles 225 Investigations 256 A Game 225 A Couple of Puzzles 257 A Sweet Trick 225 A Game 257 Technology 226 A Sweet Trick 257 Chapter Review 1 227 Technology 258 Chapter Review 2 228 Chapter Review 1 259 Chapter Review 2 260 Chapter 17 Trigonometry 2 229 Chapter 19 Data 261 A Task 229 A Task 261 History 229 History 261 Pythagoras' Theorem 230 Data 262 The Tan Ratio 231 Collecting Data 263 The Sine Ratio 232 Sampling 264 The Cos Ratio 233 Stratified Sampling 265 Trigonometry 234 Questionnaires 266 Mental Computation 238 Mental Computation 268 Competition Questions 239 Competition Questions 269 A Couple of Puzzles 240 Investigations 270 A Game 240 Technology 271 A Sweet Trick 240 A Couple of Puzzles 272 Investigations 241 A Game 272 Technology 242 A Sweet Trick 272 Chapter Review 1 243 Chapter Review 1 273 Chapter Review 2 244 Chapter Review 2 274 Paradoxes 283 Isometric Drawing 284 Chapter 20 Review 4 275 Review 1 276 Review 2 279 Glossary 285 Answers 295 Index 311 Preface This text has been written for Year 9 students. The aim of the text is to assist students in investigating and understanding the exciting and very important world of Mathematics and to implement the intent of the Australian Mathematics Curriculum. A literature review of learning from school textbooks was used to enhance the format of this textbook. Each chapter, apart from Review, contains: Numerous worked examples Numerous sets of graded exercises An open-ended rich task Mental computation Technology in mathematics Investigations Puzzles NAPLAN questions Maths competition preparation A mathematics game A mathematics trick A bit of mathematics history Careers using mathematics Chapter review Acknowledgments A heart-felt thank you to my wife Karen for your encouragement, advice, text design, images, illustrations, and above all, your loving support. Public Domain Images: pp. 17,40,48,73,143,175,189,213,229,245,261. Public Domain Images courtesy of nasaimages.org: pp. 151,156. Resources www.drdwyer.com.au Number and Algebra → Real Numbers Apply index laws to numerical expressions with integer indices. Connect different strategies for simplifying expressions with indices to illustrate the meaning of negative indices. Move fluently between representations of numeric and algebraic terms with negative indices. Apply knowledge of index laws to algebraic terms and simplify algebraic expressions, using both positive and negative integral indices. A TASK Infinite, adj. 1. exceedingly great, unlimited, immeasurably • Everyone should know about infinity. large. • Every child learns to count to ten, then to twenty, then to one hundred. • Then what? Infinity of course. Research infinity and then give a five minute explanation of infinity to the rest of the class (Try to include one of the interesting stories about infinity). A LITTLE BIT OF HISTORY A 1 m cubic block will have 1 million 1 cm cubes. • The Romans used "Decies centena milia (ten hundred thousand) as the Roman words for 1 000 000. • The French, in the 13th century, were the first to use the word 1 million. • The Indians were the most advanced with 1 large numbers and had by the 7th century =∞ defined infinity as having a denominator of 0 zero. 1 1 1 =10 =1000 =1000000 0.1 0.001 0.000001 1